Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.2% → 99.4%

Time: 7.6s

Alternatives: 10

Speedup: 21.8×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Initial Program: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(u \cdot -4\right) \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (log1p (* u -4.0)) (- s)))

C

float code(float s, float u) {
	return log1pf((u * -4.0f)) * -s;
}

Julia

function code(s, u)
	return Float32(log1p(Float32(u * Float32(-4.0))) * Float32(-s))
end

Derivation

1. Initial program 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Step-by-step derivation

   1. *-commutative 60.7%

      \[\leadsto \color{blue}{\log \left(\frac{1}{1 - 4 \cdot u}\right) \cdot s} \]

   2. log-rec 63.3%

      \[\leadsto \color{blue}{\left(-\log \left(1 - 4 \cdot u\right)\right)} \cdot s \]

   3. distribute-lft-neg-out 63.3%

      \[\leadsto \color{blue}{-\log \left(1 - 4 \cdot u\right) \cdot s} \]

   4. distribute-rgt-neg-in 63.3%

      \[\leadsto \color{blue}{\log \left(1 - 4 \cdot u\right) \cdot \left(-s\right)} \]

   5. sub-neg 63.3%

      \[\leadsto \log \color{blue}{\left(1 + \left(-4 \cdot u\right)\right)} \cdot \left(-s\right) \]

   6. log1p-def 99.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-4 \cdot u\right)} \cdot \left(-s\right) \]

   7. *-commutative 99.3%

      \[\leadsto \mathsf{log1p}\left(-\color{blue}{u \cdot 4}\right) \cdot \left(-s\right) \]

   8. distribute-rgt-neg-in 99.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{u \cdot \left(-4\right)}\right) \cdot \left(-s\right) \]

   9. metadata-eval 99.3%

      \[\leadsto \mathsf{log1p}\left(u \cdot \color{blue}{-4}\right) \cdot \left(-s\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(u \cdot -4\right) \cdot \left(-s\right)} \]

4. Final simplification 99.3%

   \[\leadsto \mathsf{log1p}\left(u \cdot -4\right) \cdot \left(-s\right) \]

Alternative 2: 93.0% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(4 + u \cdot \left(8 - u \cdot \left(u \cdot -64 - 21.333333333333332\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* u (+ 4.0 (* u (- 8.0 (* u (- (* u -64.0) 21.333333333333332))))))))

C

float code(float s, float u) {
	return s * (u * (4.0f + (u * (8.0f - (u * ((u * -64.0f) - 21.333333333333332f))))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (u * (4.0e0 + (u * (8.0e0 - (u * ((u * (-64.0e0)) - 21.333333333333332e0))))))
end function

Julia

function code(s, u)
	return Float32(s * Float32(u * Float32(Float32(4.0) + Float32(u * Float32(Float32(8.0) - Float32(u * Float32(Float32(u * Float32(-64.0)) - Float32(21.333333333333332))))))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * (single(4.0) + (u * (single(8.0) - (u * ((u * single(-64.0)) - single(21.333333333333332)))))));
end

Derivation

1. Initial program 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
2. Step-by-step derivation

   1. log-rec 63.3%

      \[\leadsto s \cdot \color{blue}{\left(-\log \left(1 - 4 \cdot u\right)\right)} \]

   2. cancel-sign-sub-inv 63.3%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(1 + \left(-4\right) \cdot u\right)}\right) \]

   3. metadata-eval 63.3%

      \[\leadsto s \cdot \left(-\log \left(1 + \color{blue}{-4} \cdot u\right)\right) \]

3. Simplified 63.3%

   \[\leadsto \color{blue}{s \cdot \left(-\log \left(1 + -4 \cdot u\right)\right)} \]

4. Taylor expanded in u around 0 93.8%

   \[\leadsto s \cdot \left(-\color{blue}{\left(-4 \cdot u + \left(-8 \cdot {u}^{2} + \left(-64 \cdot {u}^{4} + -21.333333333333332 \cdot {u}^{3}\right)\right)\right)}\right) \]
5. Step-by-step derivation

   1. *-commutative 93.8%

      \[\leadsto s \cdot \left(-\left(\color{blue}{u \cdot -4} + \left(-8 \cdot {u}^{2} + \left(-64 \cdot {u}^{4} + -21.333333333333332 \cdot {u}^{3}\right)\right)\right)\right) \]

   2. fma-def 93.8%

      \[\leadsto s \cdot \left(-\color{blue}{\mathsf{fma}\left(u, -4, -8 \cdot {u}^{2} + \left(-64 \cdot {u}^{4} + -21.333333333333332 \cdot {u}^{3}\right)\right)}\right) \]

   3. *-commutative 93.8%

      \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, \color{blue}{{u}^{2} \cdot -8} + \left(-64 \cdot {u}^{4} + -21.333333333333332 \cdot {u}^{3}\right)\right)\right) \]

   4. +-commutative 93.8%

      \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, {u}^{2} \cdot -8 + \color{blue}{\left(-21.333333333333332 \cdot {u}^{3} + -64 \cdot {u}^{4}\right)}\right)\right) \]

   5. cube-mult 93.8%

      \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, {u}^{2} \cdot -8 + \left(-21.333333333333332 \cdot \color{blue}{\left(u \cdot \left(u \cdot u\right)\right)} + -64 \cdot {u}^{4}\right)\right)\right) \]

   6. unpow2 93.8%

      \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, {u}^{2} \cdot -8 + \left(-21.333333333333332 \cdot \left(u \cdot \color{blue}{{u}^{2}}\right) + -64 \cdot {u}^{4}\right)\right)\right) \]

   7. associate-*r* 93.8%

      \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, {u}^{2} \cdot -8 + \left(\color{blue}{\left(-21.333333333333332 \cdot u\right) \cdot {u}^{2}} + -64 \cdot {u}^{4}\right)\right)\right) \]

   8. metadata-eval 93.8%

      \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, {u}^{2} \cdot -8 + \left(\left(-21.333333333333332 \cdot u\right) \cdot {u}^{2} + -64 \cdot {u}^{\color{blue}{\left(2 \cdot 2\right)}}\right)\right)\right) \]

   9. pow-sqr 93.8%

      \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, {u}^{2} \cdot -8 + \left(\left(-21.333333333333332 \cdot u\right) \cdot {u}^{2} + -64 \cdot \color{blue}{\left({u}^{2} \cdot {u}^{2}\right)}\right)\right)\right) \]

   10. associate-*r* 93.8%

       \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, {u}^{2} \cdot -8 + \left(\left(-21.333333333333332 \cdot u\right) \cdot {u}^{2} + \color{blue}{\left(-64 \cdot {u}^{2}\right) \cdot {u}^{2}}\right)\right)\right) \]

   11. distribute-rgt-out 93.8%

       \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, {u}^{2} \cdot -8 + \color{blue}{{u}^{2} \cdot \left(-21.333333333333332 \cdot u + -64 \cdot {u}^{2}\right)}\right)\right) \]

   12. distribute-lft-out 93.8%

       \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, \color{blue}{{u}^{2} \cdot \left(-8 + \left(-21.333333333333332 \cdot u + -64 \cdot {u}^{2}\right)\right)}\right)\right) \]

   13. unpow2 93.8%

       \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, \color{blue}{\left(u \cdot u\right)} \cdot \left(-8 + \left(-21.333333333333332 \cdot u + -64 \cdot {u}^{2}\right)\right)\right)\right) \]

   14. *-commutative 93.8%

       \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, \left(u \cdot u\right) \cdot \left(-8 + \left(\color{blue}{u \cdot -21.333333333333332} + -64 \cdot {u}^{2}\right)\right)\right)\right) \]

   15. *-commutative 93.8%

       \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, \left(u \cdot u\right) \cdot \left(-8 + \left(u \cdot -21.333333333333332 + \color{blue}{{u}^{2} \cdot -64}\right)\right)\right)\right) \]

   16. unpow2 93.8%

       \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, \left(u \cdot u\right) \cdot \left(-8 + \left(u \cdot -21.333333333333332 + \color{blue}{\left(u \cdot u\right)} \cdot -64\right)\right)\right)\right) \]

   17. associate-*l* 93.8%

       \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, \left(u \cdot u\right) \cdot \left(-8 + \left(u \cdot -21.333333333333332 + \color{blue}{u \cdot \left(u \cdot -64\right)}\right)\right)\right)\right) \]

   18. distribute-lft-out 93.8%

       \[\leadsto s \cdot \left(-\mathsf{fma}\left(u, -4, \left(u \cdot u\right) \cdot \left(-8 + \color{blue}{u \cdot \left(-21.333333333333332 + u \cdot -64\right)}\right)\right)\right) \]

6. Simplified 93.8%

   \[\leadsto s \cdot \left(-\color{blue}{\mathsf{fma}\left(u, -4, \left(u \cdot u\right) \cdot \left(-8 + u \cdot \left(-21.333333333333332 + u \cdot -64\right)\right)\right)}\right) \]

7. Taylor expanded in s around 0 93.8%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \left(-4 \cdot u + \left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot {u}^{2}\right)\right)} \]
8. Step-by-step derivation

   1. associate-*r* 93.8%

      \[\leadsto \color{blue}{\left(-1 \cdot s\right) \cdot \left(-4 \cdot u + \left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot {u}^{2}\right)} \]

   2. neg-mul-1 93.8%

      \[\leadsto \color{blue}{\left(-s\right)} \cdot \left(-4 \cdot u + \left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot {u}^{2}\right) \]

   3. *-commutative 93.8%

      \[\leadsto \left(-s\right) \cdot \left(\color{blue}{u \cdot -4} + \left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot {u}^{2}\right) \]

   4. unpow2 93.8%

      \[\leadsto \left(-s\right) \cdot \left(u \cdot -4 + \left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot \color{blue}{\left(u \cdot u\right)}\right) \]

   5. *-commutative 93.8%

      \[\leadsto \left(-s\right) \cdot \left(u \cdot -4 + \color{blue}{\left(u \cdot u\right) \cdot \left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right)}\right) \]

   6. sub-neg 93.8%

      \[\leadsto \left(-s\right) \cdot \left(u \cdot -4 + \left(u \cdot u\right) \cdot \color{blue}{\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u + \left(-8\right)\right)}\right) \]

   7. *-commutative 93.8%

      \[\leadsto \left(-s\right) \cdot \left(u \cdot -4 + \left(u \cdot u\right) \cdot \left(\color{blue}{u \cdot \left(-64 \cdot u - 21.333333333333332\right)} + \left(-8\right)\right)\right) \]

   8. *-commutative 93.8%

      \[\leadsto \left(-s\right) \cdot \left(u \cdot -4 + \left(u \cdot u\right) \cdot \left(u \cdot \left(\color{blue}{u \cdot -64} - 21.333333333333332\right) + \left(-8\right)\right)\right) \]

   9. fma-neg 93.8%

      \[\leadsto \left(-s\right) \cdot \left(u \cdot -4 + \left(u \cdot u\right) \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, -64, -21.333333333333332\right)} + \left(-8\right)\right)\right) \]

   10. metadata-eval 93.8%

       \[\leadsto \left(-s\right) \cdot \left(u \cdot -4 + \left(u \cdot u\right) \cdot \left(u \cdot \mathsf{fma}\left(u, -64, \color{blue}{-21.333333333333332}\right) + \left(-8\right)\right)\right) \]

   11. metadata-eval 93.8%

       \[\leadsto \left(-s\right) \cdot \left(u \cdot -4 + \left(u \cdot u\right) \cdot \left(u \cdot \mathsf{fma}\left(u, -64, -21.333333333333332\right) + \color{blue}{-8}\right)\right) \]

   12. fma-udef 93.8%

       \[\leadsto \left(-s\right) \cdot \left(u \cdot -4 + \left(u \cdot u\right) \cdot \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, -64, -21.333333333333332\right), -8\right)}\right) \]

   13. associate-*r* 93.8%

       \[\leadsto \left(-s\right) \cdot \left(u \cdot -4 + \color{blue}{u \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, -64, -21.333333333333332\right), -8\right)\right)}\right) \]

   14. distribute-lft-in 93.5%

       \[\leadsto \left(-s\right) \cdot \color{blue}{\left(u \cdot \left(-4 + u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, -64, -21.333333333333332\right), -8\right)\right)\right)} \]

   15. distribute-lft-neg-in 93.5%

       \[\leadsto \color{blue}{-s \cdot \left(u \cdot \left(-4 + u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, -64, -21.333333333333332\right), -8\right)\right)\right)} \]

9. Simplified 93.1%

   \[\leadsto \color{blue}{\left(u \cdot s\right) \cdot \left(-\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -64, -21.333333333333332\right), -8\right), -4\right)\right)} \]

10. Taylor expanded in s around 0 93.5%

    \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \left(\left(\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u - 4\right) \cdot u\right)\right)} \]
11. Step-by-step derivation

    1. mul-1-neg 93.5%

       \[\leadsto \color{blue}{-s \cdot \left(\left(\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u - 4\right) \cdot u\right)} \]

    2. associate-*r* 93.5%

       \[\leadsto -\color{blue}{\left(s \cdot \left(\left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u - 4\right)\right) \cdot u} \]

    3. *-commutative 93.5%

       \[\leadsto -\left(s \cdot \left(\color{blue}{u \cdot \left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right)} - 4\right)\right) \cdot u \]

    4. *-commutative 93.5%

       \[\leadsto -\left(s \cdot \left(u \cdot \left(\color{blue}{u \cdot \left(-64 \cdot u - 21.333333333333332\right)} - 8\right) - 4\right)\right) \cdot u \]

    5. *-commutative 93.5%

       \[\leadsto -\left(s \cdot \left(u \cdot \left(u \cdot \left(\color{blue}{u \cdot -64} - 21.333333333333332\right) - 8\right) - 4\right)\right) \cdot u \]

    6. fma-neg 93.5%

       \[\leadsto -\left(s \cdot \left(u \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, -64, -21.333333333333332\right)} - 8\right) - 4\right)\right) \cdot u \]

    7. metadata-eval 93.5%

       \[\leadsto -\left(s \cdot \left(u \cdot \left(u \cdot \mathsf{fma}\left(u, -64, \color{blue}{-21.333333333333332}\right) - 8\right) - 4\right)\right) \cdot u \]

    8. fma-neg 93.5%

       \[\leadsto -\left(s \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, -64, -21.333333333333332\right), -8\right)} - 4\right)\right) \cdot u \]

    9. metadata-eval 93.5%

       \[\leadsto -\left(s \cdot \left(u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, -64, -21.333333333333332\right), \color{blue}{-8}\right) - 4\right)\right) \cdot u \]

    10. fma-neg 93.5%

        \[\leadsto -\left(s \cdot \color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -64, -21.333333333333332\right), -8\right), -4\right)}\right) \cdot u \]

    11. metadata-eval 93.5%

        \[\leadsto -\left(s \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -64, -21.333333333333332\right), -8\right), \color{blue}{-4}\right)\right) \cdot u \]

    12. *-commutative 93.5%

        \[\leadsto -\color{blue}{u \cdot \left(s \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, -64, -21.333333333333332\right), -8\right), -4\right)\right)} \]

12. Simplified 93.1%

    \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot \left(4 - u \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, -64, -21.333333333333332\right), -8\right)\right)} \]

13. Taylor expanded in s around 0 93.5%

    \[\leadsto \color{blue}{s \cdot \left(\left(4 - \left(\left(-64 \cdot u - 21.333333333333332\right) \cdot u - 8\right) \cdot u\right) \cdot u\right)} \]

14. Final simplification 93.5%

    \[\leadsto s \cdot \left(u \cdot \left(4 + u \cdot \left(8 - u \cdot \left(u \cdot -64 - 21.333333333333332\right)\right)\right)\right) \]

Alternative 3: 90.9% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(4 + \left(u \cdot 8 + u \cdot \left(u \cdot 21.333333333333332\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* u (+ 4.0 (+ (* u 8.0) (* u (* u 21.333333333333332)))))))

C

float code(float s, float u) {
	return s * (u * (4.0f + ((u * 8.0f) + (u * (u * 21.333333333333332f)))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (u * (4.0e0 + ((u * 8.0e0) + (u * (u * 21.333333333333332e0)))))
end function

Julia

function code(s, u)
	return Float32(s * Float32(u * Float32(Float32(4.0) + Float32(Float32(u * Float32(8.0)) + Float32(u * Float32(u * Float32(21.333333333333332)))))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * (single(4.0) + ((u * single(8.0)) + (u * (u * single(21.333333333333332))))));
end

Derivation

1. Initial program 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

2. Taylor expanded in u around 0 91.9%

   \[\leadsto s \cdot \color{blue}{\left(8 \cdot {u}^{2} + \left(21.333333333333332 \cdot {u}^{3} + 4 \cdot u\right)\right)} \]
3. Step-by-step derivation

   1. associate-+r+ 91.9%

      \[\leadsto s \cdot \color{blue}{\left(\left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right) + 4 \cdot u\right)} \]

   2. +-commutative 91.9%

      \[\leadsto s \cdot \color{blue}{\left(4 \cdot u + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right)} \]

   3. *-commutative 91.9%

      \[\leadsto s \cdot \left(\color{blue}{u \cdot 4} + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right) \]

   4. unpow2 91.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \left(8 \cdot \color{blue}{\left(u \cdot u\right)} + 21.333333333333332 \cdot {u}^{3}\right)\right) \]

   5. associate-*r* 91.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \left(\color{blue}{\left(8 \cdot u\right) \cdot u} + 21.333333333333332 \cdot {u}^{3}\right)\right) \]

   6. unpow3 91.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \left(\left(8 \cdot u\right) \cdot u + 21.333333333333332 \cdot \color{blue}{\left(\left(u \cdot u\right) \cdot u\right)}\right)\right) \]

   7. unpow2 91.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \left(\left(8 \cdot u\right) \cdot u + 21.333333333333332 \cdot \left(\color{blue}{{u}^{2}} \cdot u\right)\right)\right) \]

   8. associate-*r* 91.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \left(\left(8 \cdot u\right) \cdot u + \color{blue}{\left(21.333333333333332 \cdot {u}^{2}\right) \cdot u}\right)\right) \]

   9. distribute-rgt-out 91.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \color{blue}{u \cdot \left(8 \cdot u + 21.333333333333332 \cdot {u}^{2}\right)}\right) \]

   10. distribute-lft-out 91.7%

       \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + \left(8 \cdot u + 21.333333333333332 \cdot {u}^{2}\right)\right)\right)} \]

   11. unpow2 91.7%

       \[\leadsto s \cdot \left(u \cdot \left(4 + \left(8 \cdot u + 21.333333333333332 \cdot \color{blue}{\left(u \cdot u\right)}\right)\right)\right) \]

   12. associate-*r* 91.7%

       \[\leadsto s \cdot \left(u \cdot \left(4 + \left(8 \cdot u + \color{blue}{\left(21.333333333333332 \cdot u\right) \cdot u}\right)\right)\right) \]

   13. *-commutative 91.7%

       \[\leadsto s \cdot \left(u \cdot \left(4 + \left(8 \cdot u + \color{blue}{\left(u \cdot 21.333333333333332\right)} \cdot u\right)\right)\right) \]

   14. distribute-rgt-out 91.7%

       \[\leadsto s \cdot \left(u \cdot \left(4 + \color{blue}{u \cdot \left(8 + u \cdot 21.333333333333332\right)}\right)\right) \]

4. Simplified 91.7%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot 21.333333333333332\right)\right)\right)} \]
5. Step-by-step derivation

   1. distribute-lft-in 91.7%

      \[\leadsto s \cdot \left(u \cdot \left(4 + \color{blue}{\left(u \cdot 8 + u \cdot \left(u \cdot 21.333333333333332\right)\right)}\right)\right) \]

6. Applied egg-rr 91.7%

   \[\leadsto s \cdot \left(u \cdot \left(4 + \color{blue}{\left(u \cdot 8 + u \cdot \left(u \cdot 21.333333333333332\right)\right)}\right)\right) \]

7. Final simplification 91.7%

   \[\leadsto s \cdot \left(u \cdot \left(4 + \left(u \cdot 8 + u \cdot \left(u \cdot 21.333333333333332\right)\right)\right)\right) \]

Alternative 4: 90.9% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(4 + u \cdot \left(8 + u \cdot 21.333333333333332\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* u (+ 4.0 (* u (+ 8.0 (* u 21.333333333333332)))))))

C

float code(float s, float u) {
	return s * (u * (4.0f + (u * (8.0f + (u * 21.333333333333332f)))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (u * (4.0e0 + (u * (8.0e0 + (u * 21.333333333333332e0)))))
end function

Julia

function code(s, u)
	return Float32(s * Float32(u * Float32(Float32(4.0) + Float32(u * Float32(Float32(8.0) + Float32(u * Float32(21.333333333333332)))))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * (single(4.0) + (u * (single(8.0) + (u * single(21.333333333333332))))));
end

Derivation

1. Initial program 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

2. Taylor expanded in u around 0 91.9%

   \[\leadsto s \cdot \color{blue}{\left(8 \cdot {u}^{2} + \left(21.333333333333332 \cdot {u}^{3} + 4 \cdot u\right)\right)} \]
3. Step-by-step derivation

   1. associate-+r+ 91.9%

      \[\leadsto s \cdot \color{blue}{\left(\left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right) + 4 \cdot u\right)} \]

   2. +-commutative 91.9%

      \[\leadsto s \cdot \color{blue}{\left(4 \cdot u + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right)} \]

   3. *-commutative 91.9%

      \[\leadsto s \cdot \left(\color{blue}{u \cdot 4} + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right) \]

   4. unpow2 91.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \left(8 \cdot \color{blue}{\left(u \cdot u\right)} + 21.333333333333332 \cdot {u}^{3}\right)\right) \]

   5. associate-*r* 91.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \left(\color{blue}{\left(8 \cdot u\right) \cdot u} + 21.333333333333332 \cdot {u}^{3}\right)\right) \]

   6. unpow3 91.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \left(\left(8 \cdot u\right) \cdot u + 21.333333333333332 \cdot \color{blue}{\left(\left(u \cdot u\right) \cdot u\right)}\right)\right) \]

   7. unpow2 91.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \left(\left(8 \cdot u\right) \cdot u + 21.333333333333332 \cdot \left(\color{blue}{{u}^{2}} \cdot u\right)\right)\right) \]

   8. associate-*r* 91.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \left(\left(8 \cdot u\right) \cdot u + \color{blue}{\left(21.333333333333332 \cdot {u}^{2}\right) \cdot u}\right)\right) \]

   9. distribute-rgt-out 91.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \color{blue}{u \cdot \left(8 \cdot u + 21.333333333333332 \cdot {u}^{2}\right)}\right) \]

   10. distribute-lft-out 91.7%

       \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + \left(8 \cdot u + 21.333333333333332 \cdot {u}^{2}\right)\right)\right)} \]

   11. unpow2 91.7%

       \[\leadsto s \cdot \left(u \cdot \left(4 + \left(8 \cdot u + 21.333333333333332 \cdot \color{blue}{\left(u \cdot u\right)}\right)\right)\right) \]

   12. associate-*r* 91.7%

       \[\leadsto s \cdot \left(u \cdot \left(4 + \left(8 \cdot u + \color{blue}{\left(21.333333333333332 \cdot u\right) \cdot u}\right)\right)\right) \]

   13. *-commutative 91.7%

       \[\leadsto s \cdot \left(u \cdot \left(4 + \left(8 \cdot u + \color{blue}{\left(u \cdot 21.333333333333332\right)} \cdot u\right)\right)\right) \]

   14. distribute-rgt-out 91.7%

       \[\leadsto s \cdot \left(u \cdot \left(4 + \color{blue}{u \cdot \left(8 + u \cdot 21.333333333333332\right)}\right)\right) \]

4. Simplified 91.7%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot \left(8 + u \cdot 21.333333333333332\right)\right)\right)} \]

5. Final simplification 91.7%

   \[\leadsto s \cdot \left(u \cdot \left(4 + u \cdot \left(8 + u \cdot 21.333333333333332\right)\right)\right) \]

Alternative 5: 86.9% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot \left(u \cdot 8\right) + s \cdot 4\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* u (+ (* s (* u 8.0)) (* s 4.0))))

C

float code(float s, float u) {
	return u * ((s * (u * 8.0f)) + (s * 4.0f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * ((s * (u * 8.0e0)) + (s * 4.0e0))
end function

Julia

function code(s, u)
	return Float32(u * Float32(Float32(s * Float32(u * Float32(8.0))) + Float32(s * Float32(4.0))))
end

MATLAB

function tmp = code(s, u)
	tmp = u * ((s * (u * single(8.0))) + (s * single(4.0)));
end

Derivation

1. Initial program 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

2. Taylor expanded in u around 0 87.4%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right) + 8 \cdot \left(s \cdot {u}^{2}\right)} \]
3. Step-by-step derivation

   1. associate-*r* 87.7%

      \[\leadsto \color{blue}{\left(4 \cdot s\right) \cdot u} + 8 \cdot \left(s \cdot {u}^{2}\right) \]

   2. associate-*r* 87.8%

      \[\leadsto \left(4 \cdot s\right) \cdot u + \color{blue}{\left(8 \cdot s\right) \cdot {u}^{2}} \]

   3. unpow2 87.8%

      \[\leadsto \left(4 \cdot s\right) \cdot u + \left(8 \cdot s\right) \cdot \color{blue}{\left(u \cdot u\right)} \]

   4. associate-*r* 87.8%

      \[\leadsto \left(4 \cdot s\right) \cdot u + \color{blue}{\left(\left(8 \cdot s\right) \cdot u\right) \cdot u} \]

   5. distribute-rgt-out 87.9%

      \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + \left(8 \cdot s\right) \cdot u\right)} \]

   6. *-commutative 87.9%

      \[\leadsto u \cdot \left(\color{blue}{s \cdot 4} + \left(8 \cdot s\right) \cdot u\right) \]

   7. *-commutative 87.9%

      \[\leadsto u \cdot \left(s \cdot 4 + \color{blue}{\left(s \cdot 8\right)} \cdot u\right) \]

   8. associate-*l* 87.9%

      \[\leadsto u \cdot \left(s \cdot 4 + \color{blue}{s \cdot \left(8 \cdot u\right)}\right) \]

   9. distribute-lft-out 87.8%

      \[\leadsto u \cdot \color{blue}{\left(s \cdot \left(4 + 8 \cdot u\right)\right)} \]

   10. *-commutative 87.8%

       \[\leadsto u \cdot \left(s \cdot \left(4 + \color{blue}{u \cdot 8}\right)\right) \]

4. Simplified 87.8%

   \[\leadsto \color{blue}{u \cdot \left(s \cdot \left(4 + u \cdot 8\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative 87.8%

      \[\leadsto u \cdot \left(s \cdot \color{blue}{\left(u \cdot 8 + 4\right)}\right) \]

   2. distribute-rgt-in 87.9%

      \[\leadsto u \cdot \color{blue}{\left(\left(u \cdot 8\right) \cdot s + 4 \cdot s\right)} \]

   3. *-commutative 87.9%

      \[\leadsto u \cdot \left(\left(u \cdot 8\right) \cdot s + \color{blue}{s \cdot 4}\right) \]

6. Applied egg-rr 87.9%

   \[\leadsto u \cdot \color{blue}{\left(\left(u \cdot 8\right) \cdot s + s \cdot 4\right)} \]

7. Final simplification 87.9%

   \[\leadsto u \cdot \left(s \cdot \left(u \cdot 8\right) + s \cdot 4\right) \]

Alternative 6: 86.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(4 + u \cdot 8\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (* u (+ 4.0 (* u 8.0)))))

C

float code(float s, float u) {
	return s * (u * (4.0f + (u * 8.0f)));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (u * (4.0e0 + (u * 8.0e0)))
end function

Julia

function code(s, u)
	return Float32(s * Float32(u * Float32(Float32(4.0) + Float32(u * Float32(8.0)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * (single(4.0) + (u * single(8.0))));
end

Derivation

1. Initial program 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

2. Taylor expanded in u around 0 87.9%

   \[\leadsto s \cdot \color{blue}{\left(8 \cdot {u}^{2} + 4 \cdot u\right)} \]
3. Step-by-step derivation

   1. +-commutative 87.9%

      \[\leadsto s \cdot \color{blue}{\left(4 \cdot u + 8 \cdot {u}^{2}\right)} \]

   2. unpow2 87.9%

      \[\leadsto s \cdot \left(4 \cdot u + 8 \cdot \color{blue}{\left(u \cdot u\right)}\right) \]

   3. associate-*r* 87.9%

      \[\leadsto s \cdot \left(4 \cdot u + \color{blue}{\left(8 \cdot u\right) \cdot u}\right) \]

   4. distribute-rgt-out 87.8%

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + 8 \cdot u\right)\right)} \]

   5. *-commutative 87.8%

      \[\leadsto s \cdot \left(u \cdot \left(4 + \color{blue}{u \cdot 8}\right)\right) \]

4. Simplified 87.8%

   \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(4 + u \cdot 8\right)\right)} \]

5. Final simplification 87.8%

   \[\leadsto s \cdot \left(u \cdot \left(4 + u \cdot 8\right)\right) \]

Alternative 7: 86.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot \left(4 + u \cdot 8\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* u (* s (+ 4.0 (* u 8.0)))))

C

float code(float s, float u) {
	return u * (s * (4.0f + (u * 8.0f)));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * (s * (4.0e0 + (u * 8.0e0)))
end function

Julia

function code(s, u)
	return Float32(u * Float32(s * Float32(Float32(4.0) + Float32(u * Float32(8.0)))))
end

MATLAB

function tmp = code(s, u)
	tmp = u * (s * (single(4.0) + (u * single(8.0))));
end

Derivation

1. Initial program 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

2. Taylor expanded in u around 0 87.4%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right) + 8 \cdot \left(s \cdot {u}^{2}\right)} \]
3. Step-by-step derivation

   1. associate-*r* 87.7%

      \[\leadsto \color{blue}{\left(4 \cdot s\right) \cdot u} + 8 \cdot \left(s \cdot {u}^{2}\right) \]

   2. associate-*r* 87.8%

      \[\leadsto \left(4 \cdot s\right) \cdot u + \color{blue}{\left(8 \cdot s\right) \cdot {u}^{2}} \]

   3. unpow2 87.8%

      \[\leadsto \left(4 \cdot s\right) \cdot u + \left(8 \cdot s\right) \cdot \color{blue}{\left(u \cdot u\right)} \]

   4. associate-*r* 87.8%

      \[\leadsto \left(4 \cdot s\right) \cdot u + \color{blue}{\left(\left(8 \cdot s\right) \cdot u\right) \cdot u} \]

   5. distribute-rgt-out 87.9%

      \[\leadsto \color{blue}{u \cdot \left(4 \cdot s + \left(8 \cdot s\right) \cdot u\right)} \]

   6. *-commutative 87.9%

      \[\leadsto u \cdot \left(\color{blue}{s \cdot 4} + \left(8 \cdot s\right) \cdot u\right) \]

   7. *-commutative 87.9%

      \[\leadsto u \cdot \left(s \cdot 4 + \color{blue}{\left(s \cdot 8\right)} \cdot u\right) \]

   8. associate-*l* 87.9%

      \[\leadsto u \cdot \left(s \cdot 4 + \color{blue}{s \cdot \left(8 \cdot u\right)}\right) \]

   9. distribute-lft-out 87.8%

      \[\leadsto u \cdot \color{blue}{\left(s \cdot \left(4 + 8 \cdot u\right)\right)} \]

   10. *-commutative 87.8%

       \[\leadsto u \cdot \left(s \cdot \left(4 + \color{blue}{u \cdot 8}\right)\right) \]

4. Simplified 87.8%

   \[\leadsto \color{blue}{u \cdot \left(s \cdot \left(4 + u \cdot 8\right)\right)} \]

5. Final simplification 87.8%

   \[\leadsto u \cdot \left(s \cdot \left(4 + u \cdot 8\right)\right) \]

Alternative 8: 73.6% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(u \cdot s\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* 4.0 (* u s)))

C

float code(float s, float u) {
	return 4.0f * (u * s);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = 4.0e0 * (u * s)
end function

Julia

function code(s, u)
	return Float32(Float32(4.0) * Float32(u * s))
end

MATLAB

function tmp = code(s, u)
	tmp = single(4.0) * (u * s);
end

Derivation

1. Initial program 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

2. Taylor expanded in u around 0 74.6%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
3. Step-by-step derivation

   1. *-commutative 74.6%

      \[\leadsto 4 \cdot \color{blue}{\left(u \cdot s\right)} \]

4. Simplified 74.6%

   \[\leadsto \color{blue}{4 \cdot \left(u \cdot s\right)} \]

5. Final simplification 74.6%

   \[\leadsto 4 \cdot \left(u \cdot s\right) \]

Alternative 9: 73.8% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot 4\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* u (* s 4.0)))

C

float code(float s, float u) {
	return u * (s * 4.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * (s * 4.0e0)
end function

Julia

function code(s, u)
	return Float32(u * Float32(s * Float32(4.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = u * (s * single(4.0));
end

Derivation

1. Initial program 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

2. Taylor expanded in u around 0 74.6%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right)} \]
3. Step-by-step derivation

   1. associate-*r* 74.9%

      \[\leadsto \color{blue}{\left(4 \cdot s\right) \cdot u} \]

   2. *-commutative 74.9%

      \[\leadsto \color{blue}{u \cdot \left(4 \cdot s\right)} \]

4. Simplified 74.9%

   \[\leadsto \color{blue}{u \cdot \left(4 \cdot s\right)} \]

5. Final simplification 74.9%

   \[\leadsto u \cdot \left(s \cdot 4\right) \]

Alternative 10: 16.4% accurate, 36.3× speedup

Math

\[\begin{array}{l} \\ s \cdot 0 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s 0.0))

C

float code(float s, float u) {
	return s * 0.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * 0.0e0
end function

Julia

function code(s, u)
	return Float32(s * Float32(0.0))
end

MATLAB

function tmp = code(s, u)
	tmp = s * single(0.0);
end

Derivation

1. Initial program 60.7%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

3. Applied egg-rr 17.3%

   \[\leadsto s \cdot \color{blue}{0} \]

4. Final simplification 17.3%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023238 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))